Evolutionarily stable strategies of random games, and the vertices of random polygons
Abstract
An evolutionarily stable strategy (ESS) is an equilibrium strategy that is immune to invasions by rare alternative (``mutant'') strategies. Unlike Nash equilibria, ESS do not always exist in finite games. In this paper we address the question of what happens when the size of the game increases: does an ESS exist for ``almost every large'' game? Letting the entries in the n× n game matrix be independently randomly chosen according to a distribution F, we study the number of ESS with support of size 2. In particular, we show that, as n ∞, the probability of having such an ESS: (i) converges to 1 for distributions F with ``exponential and faster decreasing tails'' (e.g., uniform, normal, exponential); and (ii) converges to 1-1/e for distributions F with ``slower than exponential decreasing tails'' (e.g., lognormal, Pareto, Cauchy). Our results also imply that the expected number of vertices of the convex hull of n random points in the plane converges to infinity for the distributions in (i), and to 4 for the distributions in (ii).